

where \(\mathcal {V}^h_0\)  is \(\mathcal {V}^h\)  with homogeneous conditions along \(\partial \Omega _u\) , \(\Delta t\)  is the time step size, and \([\chi , p^h]_{\mathcal {F}}\)  is a weighted inner product representing the classical
two-point flux approximation (TPFA) scheme. This is introduced to allow a unified
presentation of the coupled finite element/finite volume model [1]}, [2]}, [3]}. In particular, we have:
\([\chi , p^h]_{\mathcal {F}} =\sum _{\mathcal {E}_{\text{int}}} (\chi \vert _{\varphi _L} - \chi \vert _{\varphi _K} )\Upsilon _{KL} ( p^h\vert _{ \varphi _L } - p^h\vert _{ \varphi _K } ) +\sum _{\mathcal {E}_p} \chi \vert _{ \varphi _K } \Upsilon _K p^h\vert _{ \varphi _K },\) 
